Bridges Stockholm 2018 Mathematics | Art | Music | Architecture | Education | Culture 2018 Conference Proceedings Editors

Home , Gul (design), Martin Demaine, Rhythm of Structure, Susan Goldstine, The Bridges Organization

Program Chairs

Eve Torrence Bruce Torrence Department of Mathematics Department of Mathematics Randolph-Macon College Randolph-Macon College Ashland, Virginia, USA Ashland, Virginia, USA

Short Papers Chair Workshop Papers Chair

Carlo H. Séquin Kristóf Fenyvesi Computer Science Division Department of Music, Art and Culture Studies University of California University of Jyväskylä Berkeley, USA Jyväskylä, Finland

Production Chair

Craig S. Kaplan Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, Canada

Bridges Stockholm 2018 Conference Proceedings (www.bridgesmathart.org). All rights reserved. General permission is granted to the public for non-commercial reproduction, in limited quantities, of individual articles, provided authorization is obtained from individual authors and a complete reference is given for the source. All copyrights and responsibilities for individual articles in the 2018 Conference Proceedings remain under the control of the original authors.

ISBN: 978-1-938664-27-4 ISSN: 1099-6702

Cover design: Margaret Kepner, Washington, DC, USA Bridges Organization Board of Directors

Kristóf Fenyvesi George W. Hart Department of Music, Art and Culture Studies Stony Brook University University of Jyväskylä, Finland New York, USA

Craig S. Kaplan Carlo H. Séquin Cheriton School of Computer Science Computer Science Division University of Waterloo, Ontario, Canada University of California, Berkeley, USA

Sujan Shrestha Eve Torrence Science, Information Arts & Technologies Department of Mathematics Division Randolph-Macon College, Ashland, Virginia, USA University of Baltimore, Maryland, USA

Artistic and Scientific Committee Members and Coordinators

Steve Abbott Robert Fathauer Department of Mathematics Tessellations Middlebury College, Vermont, USA Phoenix, Arizona, USA Theater Event Art Exhibition

Kristóf Fenyvesi Susan Gerofsky Department of Music, Art and Culture Studies University of British Columbia University of Jyväskylä, Finland Vancouver, Canada Family Day Theater Event

Sarah Glaz Tiffany Inglis University of Connecticut D2L Storrs, Connecticut, USA Waterloo, Ontario, Canada Poetry Reading Technical Support

Katie McCallum Nathan Selikoff University of Brighton Digital Awakening Studios Brighton, England, UK Orlando, Florida, USA Art Exhibition Technical Support

Bianca Violet IMAGINARY Berlin, Germany Short Film Festival Conference Organization

The Managing Board of the National Museum of Science and Technology Peter Skogh, Museum Director Stockholm, Sweden Mariana Back Annika Brantingson Curator Coordinator Conferences/Events National Museum of Science and Technology National Museum of Science and Technology Stockholm, Sweden Stockholm, Sweden Lars Paulsson Linda Sandberg Curator Program Manager National Museum of Science and Technology National Museum of Science and Technology Stockholm, Sweden Stockholm, Sweden Cecilia Sommer Host Manager National Museum of Science and Technology Stockholm, Sweden

Katrin Amberntsson Annika Callen Björn Camitz Strategic program coordinator Museum & Bridges Conference host Technician The National Museums of World Culture/ National Museum of Science and National Museum of Science and Ethnographic, Stockholm, Sweden Technology Technology Stockholm, Sweden Stockholm, Sweden Christopher Einarsson Christopher Eliasson Karolina Furtenbach Museum Educator Educator Communicator National Museum of Science and Police Museum National Museum of Science and Technology Stockholm, Sweden Technology Stockholm, Sweden Stockholm, Sweden Annika Gooch Sara Grane Lina Granér Program director Conference Coordinator Museum Host, Maritime Museum Museum of Performing Arts The National Museums of World Culture/ Stockholm, Sweden Stockholm, Sweden Ethnographic, Stockholm, Sweden

Kristina Hogvall Britta Isaksson-Bergholm Marko Klemetti Event Coordinator Museum Educator Maintenance Manager National Museum of Science and National Museum of Science and National Museum of Science and Technology Technology Technology Stockholm, Sweden Stockholm, Sweden Stockholm, Sweden Cecilia Kozma Kicki Miles Josefina Nilsson Head of House of Science Web-manager Communicator Royal Institute of Technology National Museum of Science and National Museum of Science and Stockholm University, Sweden Technology Technology Stockholm, Sweden Stockholm, Sweden Dan Nordström Peter Nyström Maria Olsson Technician National Center of Mathematic Project leader/ Accessibility advisor National Museum of Science and Education (NMC) National Museum of Science and Technology University of Gothenburg Technology Stockholm, Sweden Gothenburg, Sweden Stockholm, Sweden

Gunilla Stillström Anders Wallenthin Curator Museum & Bridges Conference host National Sports Museum National Museum of Science and Stockholm, Sweden Technology Stockholm, Sweden National Museum of Science and Technology Bridges Advisory Committee

Cissi Askwall Tom Callen Science for All Nordic4DFrame Stockholm, Sweden Vaxholm, Sweden Betsy Devine Lena Gumaelius Nobel Center Royal Institute of Technology Stockholm, Sweden Stockholm University, Sweden Annika Hedås Falk Ann-Catherine Fröjdå Nobel Center National Travelling Theater Stockholm, Sweden Stockholm City, Sweden Anki Hellberg Elisabeth Söder ArtsAdventure Media Center Helsinki, Finland Stockholm City, Sweden Mikal Vejdemo-Johnsson Royal Institute of Technology Stockholm, Sweden

Proceedings Program Committee

Steve Abbott Abdalla G. M. Ahmed Mara Alagic Middlebury College Khartoum, Sudan Wichita State University Vermont, USA Kansas, USA Ellie Baker Debra K. Borkovitz Robert Bosch Lexington, Massachusetts, USA Wheelock College Oberlin College Boston, Massachusetts, USA Ohio, USA Christopher Brownell Doug Burkholder Christopher Carlson Fresno Pacific University Lenoir-Rhyne University Wolfram Research California, USA Hickory, North Carolina, USA Champaign, Illinois, USA David Chappell Andrew Cooper Kelly Delp University of La Verne North Carolina State University Cornell University California, USA Raleigh, North Carolina, USA Ithaca, New York, USA Erik D. Demaine Martin Demaine Neil Dodgson Massachusetts Institute of Technology Massachusetts Institute of Technology Victoria University of Wellington Cambridge, Massachusetts, USA Cambridge, Massachusetts, USA New Zealand Carol Dorf Mircea Draghicescu Doug Dunham Berkeley High School ITSPHUN LLC University of Minnesota California, USA Portland, Oregon, USA Duluth, USA Frank Farris Robert Fathauer Kristóf Fenyvesi Santa Clara University Tessellations University of Jyväskylä California, USA Phoenix, Arizona, USA Finland James Forren Paul Gailiunas Susan Gerofsky Dalhousie University Newcastle, England University of British Columbia Halifax, Nova Scotia, Canada Vancouver, Canada Sarah Glaz Susan Goldstine Chaim Goodman-Stauss The University of Connecticut St. Mary's College of Maryland University of Arkansas Storrs, Connecticut, USA Maryland, USA Fayetteville, Arkansas, USA Gary Greenfield Emily Grosholz George Hart University of Richmond Pennsylvania State University Stony Brook University Virginia, USA University Park, Pennsylvania, USA New York, USA Andrea Hawksley Elisabeth Heathfield Judy Holdener eleVR, HARC, YCR Bluewater District School Board Kenyon College San Francisco, California, USA Ontario, Canada Ohio, USA Patrick Honner Tiffany Inglis Veronika Irvine Brooklyn Technical High School D2L University of Waterloo New York City, NY, USA Waterloo, Ontario, Canada Ontario, Canada Craig S. Kaplan Karl Kattchee Eva Knoll Cheriton School of Computer Science University of Wisconsin Mount Saint Vincent University University of Waterloo, Ontario, Canada La Crosse, USA Halifax, Nova Scotia, Canada Darci Kracht Zsolt Lavicza Kirsi Peltonen Kent State University Johannes Kepler University Aalto University Ohio, USA Linz, Austria Helsinki, Finland Peter J. Lu Penousal Machado Alice Major Harvard University University of Coimbra Edmonton, Alberta, Canada Cambridge, Massachusetts, USA Portugal Vincent J. Matsko Elisabetta Matsumoto Dan May Educational Consultant Georgia Institute of Technology Black Hills State University San Francisco, California, USA Atlanta, Georgia, USA Spearfish, South Dakota, USA Doug McKenna Kerry Mitchell Mike Naylor Mathemaesthetics, Inc. Phoenix, Arizona, USA Matematikkbølgen Boulder, Colorado, USA Math Creativity and Competency Center Vanvikan, Norway Doug Norton Osmo Pekonen Rinus Roelofs Villanova University University of Jyväskylä Hengelo, The Netherlands Pennsylvania, USA Finland Radmila Sazdanovic Karl Schaffer Henry Segerman North Carolina State University De Anza College and MoveSpeakSpin Oklahoma State University Raleigh, North Carolina, USA Scotts Valley, California Stillwater, Oklahoma, USA Carlo H. Séquin Sujan Shrestha Donald Spector University of California, Berkeley University of Baltimore Hobart & William Smith Colleges USA Maryland, USA Geneva, New York, USA Stacy Speyer David Swart John Sullivan Cubes and Things Waterloo, Ontario, Canada Technische Universität Berlin Alameda, California, USA Berlin, Germany Felicia Tabing Briony Thomas Bruce Torrence University of Southern California University of Leeds Randolph-Macon College Los Angeles, California, USA England Ashland, Virginia, USA Eve Torrence Godfried Toussaint Tom Verhoeff Randolph-Macon College New York University Eindhoven University of Technology Ashland, Virginia, USA Abu Dhabi, United Arab Emirates The Netherlands Charles Wampler Phil Webster Carolyn Yackel General Motors Research and Phil Webster Design Mercer University Development Scotts Valley, California, USA Macon, Georgia, USA Warren, Michigan, USA Rosa Zwier Melbourne, Victoria, Australia Art Exhibition and Catalog Program Committee

Robert Fathauer Katie McCallum Tessellations Company University of Brighton Phoenix, Arizona, USA England Conan Chadbourne Lars Paulsson San Antonio, Texas, USA National Museum of Science and Technology Stockholm, Sweden Kristóf Fenyvesi Nathan Selikoff University of Jyväskylä Digital Awakening Studios Jyväskylä, Finland Orlando, Florida, USA Abdalla G. M. Ahmed Karl Kattchee Khartoum, Sudan Department of Mathematics University of Wisconsin, LaCrosse, USA Gabriele Meyer Marty Levin Department of Mathematics Portland, Oregon, USA University of Wisconsin, Madison, USA

Short Film Festival Program Committee

Evelyn Lamb Francesco Mancini Salt Lake City, Utah, USA The Garden of Archimedes Museum Florence, Italy Giulio Rasi Bianca Violet Technical University IMAGINARY Berlin, Germany Berlin, Germany

Contents

Preface ...... xix

Invited Papers

Marjorie Rice and Her Pentagonal Tilings ...... 1 Doris Schattschneider

Some Memories of Koos Verhoeff (1927 – 2018)...... 3 Tom Verhoeff

A Personal Approach to the Art of Mathemagic with a Deck of Cards ...... 7 Colm Mulcahy

Regular Papers

Art and Recreational Math Based on Kite-Tiling Rosettes...... 15 Robert Fathauer

Re-Generating Continuous Rumˆı Compositions ...... 23 Nadide Ebru Yazar and Tugrul˘ Yazar

4-3 Dissection Tiling System ...... 31 Andrew Sniderman

Multi-Scale Truchet Patterns ...... 39 Christopher Carlson

Monochrome Map Weaving with Truchet-Like Tiles ...... 45 Abdalla Gaafar Merghani Ahmed

Phoenix – Symbol of Mathematics...... 53 Tuomas Nurmi

A Poor Man’s Hyperbolic Square Mapping ...... 59 Chamberlain Fong and Douglas Dunham

Kaleidoscopes for Non-Euclidean Space ...... 67 Peter Stampﬂi

ix New Metamorphosis Patterns ...... 75 Douglas Dunham and John Shier

A Minimal Art Object with Four Famous Fabulous Faces ...... 83 Walt van Ballegooijen and Hans Kuiper

Constructing 3D Perspective Anamorphosis via Surface Projection ...... 91 Tiffany Inglis

Mondrian Revisited: A Peek Into The Third Dimension ...... 99 Martin Skrodzki and Konrad Polthier

The Art and Mathematics of Self-Interlocking SL Blocks...... 107 Shen-Guan Shih

Torus and Klein Bottle Tessellations with a Single Tile of Pied de Poule (Houndstooth) ...... 115 Loe M.G. Feijs

Homage to Charles O. Perry ...... 123 Carlo H. Sequin´

Modular Toroids Constructed from Nonahedra ...... 131 Yifat Amir and Carlo H. Sequin´

Weaving Double-Layered Polyhedra...... 139 Rinus Roelofs

Two-Layer Woven Surfaces with Planar Faces...... 147 Ulrich Reitebuch, Eric Zimmermann and Konrad Polthier

Laminar Reciprocal Structures ...... 155 Javier Barrallo, Francisco Gonzalez-Quintial´ and Antonio Sanchez-Parandiet´

The Beauty of the Symmetric Sierpinski Relatives ...... 163 Tara Taylor

Sphairahedra and Three-Dimensional Fractals ...... 171 Kento Nakamura and Kazushi Ahara

Plane-Filling Folding Curves on the Square Grid ...... 179 Jorg¨ Arndt and Julia Handl

On a Better Golden Rectangle (That is Not 61.8033. . . % Useless!) ...... 187 Douglas McKenna

x Loop-Forms: From Construction to Composition ...... 195 James Mai

The Pentagram and the Golden Angle in Almada Negreiros’ Mural “Comec¸ar” ...... 203 Pedro J. Freitas and Simao˜ Palmeirim

Illustrating the Theory of Numbers ...... 211 Martin Weissman

Walking Around Trees: A 6-Letter ‘DNA’ for Baskets with Handles ...... 219 James Mallos

A New Family Satisfying the Intersection Graph Conjecture ...... 225 Ally Stacey

Textures in Simulations of Biological Cell Growth ...... 231 David Chappell

A Class of Spherical Penrose-Like Tilings with Connections to Virus Protein Patterns and Modular Sculpture ...... 237 Hamish Todd

Hex-Chaos Compositions and Equivalence Classes of Packing Problems...... 245 Gary Greenﬁeld

Combinatorial Greetings from Georges Perec ...... 253 Tatiana Bonch-Osmolovskaya

Hyperspace, Poetic Science Fiction and Algebraic Topology ...... 259 Emily Grosholz

Geometry of Quadrangles in Almqvist’s “The Queen’s Tiara” ...... 265 Tiina Katriina Kukkonen

The Mathematical Center of Attention, its Attributes and Motion Analyses in Dance Choreography...... 273 Karl Schaffer, Joseph Thie and Kasia Williams

The Mathematical Circus Project ...... 281 Andreia Hall and Sonia´ Pais

Music from Vibrating Wallpaper ...... 287 Frank Farris

xi The Checkerboard of Tunes ...... 295 Andrew Crompton

A Generalized Dual of the Tonnetz for Seventh Chords: Mathematical, Computational and Compositional Aspects...... 301 Sonia Cannas and Moreno Andreatta

From the Cheesegrater to the Parthenon: A Musical Odyssey ...... 309 Terry Trickett

Math in the Studio...... 317 Judy Holdener and Karen Snouffer

Catenary Arch Constructions ...... 325 George Hart and Elisabeth Heathﬁeld

Teaching Mathematics and Physics for Animation in Processing ...... 333 Lali Barriere`

When Mathematics Meets Art: How Might Art Contribute to the Understanding of Mathematical Concepts? ...... 341 Liora Nutov

Short Papers

Dance Art, Math, Education – an Eternal Triangle...... 347 Paul Moerman

Turning Math into Dance: Lessons from Dancing My PhD ...... 351 Nancy Scherich

Voronoi Diagrams: Didactical and Artistic Applications ...... 355 Sandra Bento, Helena Ferreira and Andreia Hall

Play Truchet: Using the Truchet Tiling to Engage the Public with Mathematics ...... 359 Cindy Lawrence

Visualizing Symmetry Subgroup Structures Using Simple Motifs...... 363 David Reimann

The Marvellous Bridging of Maths and Art Education and its Relation to Cognitive and Emotional Development ...... 367 Gunnel Berlin

xii Don’t Preach Facts – Stimulate Acts ...... 371 Peter Baptist and Carsten Miller

Geometry and Origami to Share Cultural Heritage: Results of the Experimentation “The King and the Origami” at the Royal Residence of Venaria ...... 375 Paolo Armand, Caterina Cumino, Martino Pavignano, Maria Luisa Spreaﬁco and Ursula Zich

Developing Mathematical and Technological Competencies of Students Through Remodeling Games and Puzzles ...... 379 Diego Lieban, Marina Menna Barreto, Sandra Reichenberger, Zsolt Lavicza and Ruana M. Schneider

Science Spaces: An Open Workshop Concept to Create Science Exhibits ...... 383 Bianca Violet and Milena Damrau

Musical Scales and Multiplicative Groups ...... 387 Donald Spector

Rock Me Fibonacci: Using Recurrence Relations and State-Transition Matrices to Count Rock Drum Fill Patterns ...... 391 Joshua Holden

Combinatorics, Probability and Choice in Music Composition: Towards an Aesthetics of Composing Systems for Non-Musicians...... 395 Giovanni Albini

‘Dreamlinks’: Link Theory Meets Music Composition. An Introduction to Compositional Methods Related to Primary Links ...... 399 Saverio Tesolato

Just Intonation Keyboard: Isomorphic Keyboard Reimagined ...... 403 Marek Zabkaˇ

The Area Set Value Relationships in Atonal Music ...... 407 Nikita Mamedov

Landmarks in Algebra Quilt ...... 411 Elaine Ellison

Algoritmisch Ritme: Algorithmic Art as Material in an Interactive Dance-Projection ...... 415 Daphne Muller and Loe M.G. Feijs

xiii A Methodology of Leaping Iteration for Drawing ...... 419 Ming Jang Chen

Surﬁng the Mobius¨ Band: An Example of the Union of Art and Mathematics ...... 423 Francisco Saez de Adana

Capturing the Visual Traits of a Mathematician – On Anders Johan Lexell’s Futile Studies in Physiognomy ...... 427 Johan Sten´ and Martina Reuter

Representing the Undecidable ...... 431 Michel Tombroff

Observations on Concave Perspective ...... 435 Stephen Campbell

Which Integer Is the Most Mysterious? ...... 439 Osmo Pekonen

Mapping from e to Metaphor ...... 443 Alice Major

Art of de Bruijn Sequences ...... 447 Karl Kattchee

Card Shufﬂing Visualizations ...... 451 Roger Antonsen

The Art and Mathematics of Cycling: Using Old Bicycles to Draw Spirograph Patterns ...... 455 Nick Sayers

Juan Sanchez´ Cotan:´ Seeing Mathematically ...... 459 Paul Zorn

Mathematics in Drafting Japanese Crest Designs ...... 463 Felicia Tabing

Glass Mosaics Using Right-Triangle Subdivision ...... 467 Thomas Denker

A Plethora of Patterns from Discrete Spiraled Sequences ...... 471 Tom Bates

xiv Computationally Intensive Puns with Figurative Subgraphs...... 475 Robert Bosch

To the World’s End/ A Circle Bundle Over a Circle ...... 479 Zachary Treisman and Lun-Yi Tsai

Folding Space-Filling Bisymmetric Hendecahedron for a Large-Scale Art Installation ...... 483 Jiangmei Wu and Guy Inchbald

A Threefold Mobius¨ Band with Constant Twist and Minimal Bending as the Limit of Tetrahedral Rings ...... 487 Johannes Schonke,¨ Michael Grunwald and Eliot Fried

3D Printing Chaos ...... 491 Micahel Gagliardo

Compound Parallelohedra Building Blocks with Creature-Like Morphologies ...... 495 Akihiro Matsuura

Method for Designing a Hinged Cube Puzzle...... 499 Chirag Mehta

A Design Method Based on Close-Packing Circles and Spheres of Multiple Sizes for Designers and Architects ...... 503 Roger Burrows

Knot Designs Based on Rhombille Tiling Notations ...... 507 Nithikul Nimkulrat and Tuomas Nurmi

Girih Tiles in 3D ...... 511 Ulrich Reitebuch, Henriette-Sophie Lipschutz¨ and Konrad Polthier

Origami Explorations of Convex Uniform Tilings Through the Lens of Ron Resch’s Linear Flower ...... 515 Uyen Nguyen and Ben Fritzson

Self-Diagramming Lace...... 519 Susan Goldstine

Mobius¨ Cellular Automata Scarves...... 523 Elisabetta Matsumoto, Henry Segerman and Fabienne Serriere

Lizardy Loops: The Savvy Selection of Sinuous Sequences of Circular Sectors ...... 527 Phil Webster

xv Horosphere, Cyclide and 3d Hyperbolic Tilings ...... 531 Vladimir Bulatov

Folding the Vesica Piscis ...... 535 Klara Mundilova and Tony Wills

Realizations and Constructions of Minimally Rigid Graphs ...... 539 Georg Grasegger

Polyhedral Models of the Projective Plane ...... 543 Paul Gailiunas

Extending Mandelbox Fractals with Shape Inversions ...... 547 Gregg Helt

Virtual Crocheting of Euclidean Planes in a 3-Sphere ...... 551 Eryk Kopczynski´ and Dorota Celinska´

Designing Beaded Sculptures Inspired by Clathrate Hydrates ...... 555 Yuan-Jia Fan, Bih-Yaw Jin and Chia-Chin Tsoo

Desargues Conﬁguration as a Gnomonic Projection ...... 559 Taneli Luotoniemi

Action Modular Origami ...... 563 Tung Ken Lam

Workshop Papers

Sevenfold and Ninefold Mobius¨ Kaleidocycles...... 567 Michael Grunwald, Johannes Schonke¨ and Eliot Fried

Self-Similarity and the Tumbling Square ...... 575 Susan Happersett

Adopt a Polyhedron – A Citizen Art Project in Mathematics...... 579 Anna Maria Hartkopf and Gunter¨ M. Ziegler

Folding Curlicue and Exploring Its Mathematical Properties ...... 585 Natalija Budinski

Geometry and Origami to Make Dynamic Street Art ...... 589 Sara De Grandis, Silvia Fiore, Maria Luisa Spreaﬁco, Marco Torredimare, Margherita Truffa and Ursula Zich

xvi Geometrical Object Making For Design Thinking...... 597 Gizem Aytac¸

Open Geoboard – a Platform for Art, Math and Inspiration...... 603 Yordan Hodzhev and Nikola Chernev

Creating Painting Puzzles: Math, Art, Games and Technology ...... 609 Marina Menna Barreto, Diego Lieban and Barbara Kimeswenger

Constructing Linkages for Drawing Curves ...... 613 Barbara Kimeswenger, Georg Grasegger and Sandra Reichenberger

A Hands-on Laboratory with Mathematical Mechanical Drawing Machines ...... 617 Laura Farroni and Paola Magrone

Renewable Energy Resources for Mathematics Learning: Windmills and Water Wheels at the Math Class ...... 623 Kristof´ Fenyvesi, Ho-Gul Park, Kwang-Cheol Song, Zsolt Lavicza and Mariana Back

Engaging Community Through the Integration of Art and Mathematics ...... 629 Ellie Balk and Tricia Stanley

Let’s Sketch in 360o: Spherical Perspectives for Virtual Reality Panoramas ...... 637 Antonio´ Araujo´

Poetry Puzzles ...... 645 Lisa Lajeunesse

The Theory-Headed Poem ...... 649 Carol Dorf

Reimagining the Mathematical Paper ...... 651 Katie McCallum

Experiencing Group Structure: Observing, Creating and Performing the Plain Hunt on 4 via Music, Poetry, Visual and Culinary Arts ...... 659 Susan Gerofsky, Eva Knoll, Tara Taylor and Avalon Campbell-Cousins

Exploring the Geometry of Music with Technology: Rhythm, Scales and Temperament via Geogebra ...... 667 Steven Bleiler, Ewan Kummel and Michael Wells

Using Math to Create a Musical Sandbox ...... 673 Esmee Verschoor, Alyssa Eggersgluss, Collin Goldbach, Annmarie Thomas and Allison Knoph

xvii Human Encryptable Visual Cryptography...... 675 Andrea Hawksley and Andrew Lutomirski

Mathematical Magic With a Deck of Cards...... 681 Jorge Nuno Silva, Pedro J. Freitas and Tiago Hirth

Author Index...... 687

xviii Preface

Welcome to the 21st Bridges conference! This year the conference is being held in Stockholm, Sweden at the National Museum of Science and Technology (Tekniska museet). Stockholm is known for its ancient palaces, cathedrals, and cobbled lanes, and boasts one of Europe’s best-preserved medieval city centers. Stockholm is also a dynamic modern city which is famous for its higher learning and scientific research institutions, such as the Royal Institute of Technology, Karolinska Institute, and the Nobel Center, as well as its museums, theaters, art galleries and concert venues. Located beside an archipelago of over 30,000 islands and containing a large number of parks, Stockholm is one of the greenest cities in the world and a particularly lovely city for enjoying the long twilight summer evenings found at this high northern latitude. The National Museum of Science and Technology was one of the first museums of its kind when it opened in 1936. The museum attracts over 300,000 visitors every year and is the home of Sweden’s first science center, which was completely renovated into the highly interactive learning environment, MegaMind, in 2015. The museum was named the Swedish Museum of the Year in 2016 and received the international Children in Museums Award for MegaMind in 2017. The museum’s Mathematical Garden, which opened in autumn 2017, is an accessible playground that covers the entire area in front of the museum. The garden’s five zones are each based on a different mathematical theme. The center of the garden features a tessellation by the U.S. amateur mathematician Majorie Rice on the ramp leading to the museum’s entrance. With the opening of MegaMind and the Mathematical Garden the museum has undergone a transformation from a more traditional technology museum into an arena of discovery, with a clear focus on children and their families. This year’s Bridges Program Co-chairs are Eve and Bruce Torrence. They coordinated an international Program Committee of over 70 experts who provided extensive reviews and editorial comments on submissions. They also served as chairs of the regular paper track. Carlo Séquin chaired the short papers track and Kristóf Fenyvesi chaired the workshop submissions. Many thanks go to the members of the Program Committee who reviewed the large number of papers received. Special thanks to Bianca Violet, our Short Film Festival Chair; Evelyn Lamb, Giulio Rasi, and Francesco Mancini, who served as the jury for the Film Festival; Sarah Glaz, the Poetry Reading Chair; Steve Abbott and Susan Gerofsky, the Theater Event Chairs; Kristóf Fenyvesi for organizing the public Family Day; Margaret Kepner for designing our beautiful book covers and Phil Webster for the 2018 postcard design. This year for the first time, the conference will include a Mathematical Fashion Show, organized by Susan Goldstine, on the evening of Informal Music Night, which is hosted by Mike Naylor. The 2018 edition of the Bridges proceedings includes 3 invited papers, 44 regular papers, 55 short papers, and 21 workshop papers. You will find new work on using the arts to explore topics in tiling, cellular automata, fractals, knot theory, number theory, polyhedra, topology, space-filling curves, as well as mathematical concepts in dance, music, biology, chemistry, architecture and art history. There are papers on mathematics in literature, including historical novels, poetry, and comic books. Other fascinating papers explore card shuffling patterns and using 3D printing to present chaos, musical concepts, and portraits. There is even a witty discourse on the golden rectangle. Fiber arts are well represented with papers on weaving, quilting, lacemaking, knitting, crochet, and beading. Examples of public outreach through dance and magic performances, a mathematical circus, large scale art installations, a bicycle parts spirograph, and street art are explored as well as ways to engage students of all ages in exciting classroom activities. Particularly intriguing are two papers by graduate students in mathematics who used painting and dance to understand and express patterns related to concepts in their dissertations. Thank you to all of the authors and reviewers for their generous contributions to this year’s proceedings, with special thanks to David Swart, George Hart, and Craig S. Kaplan for their endless advice and support. This year we were able to offer $10,000 USD in student travel scholarships to 14 students from around the world who authored accepted papers and created mathematical artworks. We are very grateful

xix to Jade Vinson for his generous contribution, which made this program possible, and to Sujan Shrestha who managed the program. An exhibition of mathematical art has been an annual feature of Bridges since 2001. Artists from Europe, Africa, Asia, and North and South America will be represented. A wide variety of artistic media are included in the exhibition, including 2D and 3D digital prints, painting, beadwork, ceramics, wood, metal, quilting, and paper folding. Artists drew inspiration from the mathematics of fractals, polyhedra, non-Euclidean and four-dimensional geometry, tiling, knot theory, number theory, and more. This year Katie McCallum and Robert Fathauer served as co-curators of the exhibition. The jury considering the artworks consisted of Abdalla Ahmed, Karl Kattchee, Martin Levin and Gabriela Meyer. The print catalog was prepared by Conan Chadbourne with cover art by Margaret Kepner, and the art submission website was created and administered by Nathan Selikoff. Lars Paulsson, the Curator/Project Leader for the Mathematical Garden, was the local coordinator in Stockholm for the art exhibition. Many people contributed to the organization of this year’s conference. A debt of gratitude and thanks to Museum Director Peter Skogh, who supported the conference planning throughout its various phases. We would also like to thank the entire Museum’s Board of Directors for their enthusiasm for the project. Special thanks to Mariana Back, General Chair for Bridges 2018. Mariana is a Curator and Development Leader for the Museum’s highly interactive science center, MegaMind. She took the initiative to have Bridges 2018 in Stockholm after participating in Bridges 2016 in Jyväskylä, Finland. Following that meeting, she realized that the combination of MegaMind and the recently opened Mathematical Garden would make the National Museum of Science and Technology an interesting venue for a Bridges conference. Kristóf Fenyvesi provided Mariana with endless encouragement, without which she would not have taken the idea any further. Event Managers Annika Brantingson and Kristina Hogvall have been a tremendous support during the planning process as well as during the conference. Staff Manager Cecilia Sommer, together with Head Technician Marko Klemetti, Conference Technician Björn Camitz, Conference Hosts Annika Callen and Anders Wallenthin, and Museum Educators Britta Isaksson- Bergholm and Christopher Einarsson strove to ensure everything would run smoothly during the conference. Program Manager Linda Sandberg and Communication and Marketing’s Josefina Larsson and Karolina Furtenbach all played an important role in the local project group. Sebastian Flavet had the social media channels under control throughout the conference. We would also like to thank the National Museum of Science and Technology’s local Advisors: Annika Hedås Falk and Betsy Devine from the Nobel Center, Stockholm; Elisabeth Söder from the Media Center, Stockholm City; Tom Callen of Nordic4DFrame AB, Vaxholm; Mikal Vejdemo-Johnsson from the Royal Institute of Technology, Stockholm; Lena Gumaelius from Stockholm University/Royal Institute of Technology; Cissi Askwall of Science for All (Vetenskap & Allmänhet), Stockholm; Anki Hellberg from ArtsAdventure, Helsinki; and Ann-Catherine Fröjdå of the National Touring Theater, Stockholm. Visit Stockholm has been most helpful, and we would also like to thank the publishing company, Natur och kultur, for support in connection to activities in the Mathematical Garden. We gratefully acknowledge the support of our sponsors in making Bridges 2018 possible. Heartfelt thanks to the Foundation for the National Museum of Science and Technology's 1974 Jubilee Fund, the Ethnographic Museum, the Maritime Museum/Sjöhistoriska, the Swedish Sports Museum, the Police Museum, and the Swedish Museum of Performing Arts. Many thanks to the Royal Institute of Technology, Stockholm University, the National Center of Mathematics Education (NCM), the Nobel Museum and Nobel Center, the National Touring Theater, ArtsAdventure, Swedish National Heritage Board and many others who helped with advice and to spread information about the Conference. We hope you enjoy the 2018 Bridges Conference and this wonderful collection of new ideas. The joyful and creative spirit of our conference’s founder, Reza Sarhangi, lives on in Bridges and continues to bring inspiration to mathematicians and artists around the world.

The Bridges Organization Board of Directors www.bridgesmathart.org